Electrical circuits typically are comprised of linear, passive circuit elements, such as resistors, inductors, and capacitors as well as non-linear and active circuit elements, such as transistors and power sources. In the process of designing circuits, particularly large-scale integrated circuits, it is normal practice to mathematically model the circuit to simulate performance. Particularly, the outputs of the circuit are modeled as a function of the inputs of the circuit. The mathematical model is used to determine various response characteristics of the circuit.
In our prior U.S. patent application Ser. No. 08/269,230, filed Jun. 30, 1994, now U.S. Pat. No. 5,537,329, Ser. No. 08/489,270, filed Jun. 9, 1995, now U.S. Pat. No. 5,689,685, and Ser. No. 08/904,233, filed Jul. 31, 1997, all of which are incorporated herein by reference, we disclosed new methods and apparatus for modeling linear subcircuits within a larger electrical circuit.
An electrical circuit, such as an integrated circuit, frequently contains large portions thereof comprised entirely of linear circuit elements. For instance, a wire or other connector usually can be modeled as a network of resistances and capacitances. Accordingly, the entire interconnect structure between a last non-linear circuit element in an integrated circuit and the end of a package pin can be mathematically modeled as a linear subcircuit.
The modeling of the larger overall circuit can be extremely complex in large scale integrated (LSI) circuits and very large scale integrated (VLSI) circuits. Our aforementioned patent applications disclose techniques for modeling the linear subcircuit portions within the larger circuit by a greatly reduced model compared to the equations describing the linear subcircuit.
Referring to our aforementioned patent applications, in U.S. patent application Ser. No. 08/269,230, we disclosed a Pade via Lanczos method for determining a scalar Pade approximant of a transfer function of a linear circuit for use in determining, with substantial accuracy, the frequency response behavior of the linear circuit.
In U.S. patent application Ser. No. 08/489,270, we disclosed an extension to the previous application in which the Pade via Lanczos method was extended to matrix transfer functions. More specifically, U.S. patent application Ser. No. 08/489,270 discloses a numerically stable method to determine a matrix Pade approximation of the frequency response of a linear circuit or subcircuit with little numerical degradation, and providing an acceptable computational cost per order of approximation. The method reduces the very large matrices used to represent the p by p matrix transfer function to a pair of much smaller matrices so that the resulting approximate matrix transfer function has basically the same characteristics as the original matrix transfer function. Using the method and apparatus disclosed therein, the original matrix can be reduced to any particular size desired. The accuracy of the approximation depends on the order of reduction in size of the matrices. Generally, the matrices can be reduced to a very large extent without significant decrease in accuracy.
In U.S. patent application Ser. No. 08/904,223, we disclosed a specialized adaptation of the method and apparatus of U.S. patent application Ser. No. 08/489,270, discussed above, for use in connection with passive, linear circuits or subcircuits in which the matrices are guaranteed to be symmetric. That method can reduce circuit simulation processing load by as much as one half for passive linear circuits. Further, when the additional condition is met that the circuit contains only two types of linear circuit elements, the matrices are guaranteed to be not only symmetric, but also positive definite, which guarantees stability and passivity of the approximation.
In the method and apparatus of application Ser. No. 08/904,223, however, linear circuits that contain R, L and C circuit elements as well as linear circuits with different input and output vectors are not guaranteed to result in a symmetric Lanczos matrix. Therefore, they are not guaranteed to result in a stable and passive Pade approximate transfer function. However, if the original circuit is stable and possibly passive, then, in some applications, it is crucial that the reduced-order models also have these properties.
Therefore, it is an object of the present invention to provide an improved method and apparatus for generating a Pade approximant of a transfer function of a linear circuit.
It is a further object of the present invention to provide an improved method and apparatus for generating a partial Pade approximant of a transfer function of a linear circuit that is passive and stable.
It is another object of the present invention to provide an improved method and apparatus for modifying a Lanczos matrix corresponding to a Pade approximant of a transfer function of a linear circuit that is not passive and stable so as to make it passive and stable with as minimal a sacrifice in the accuracy of the approximant as possible.